If `vecA+vecB=vecR` and `2vecA+vecB` is perpendicular to `vecB` then

Options

A. A=R
B. B=2R
C. B=R
D. B=A

Explanation

To solve the problem, we start with the given equations and conditions:

1. **Given Equations**:
- \(\vec{A} + \vec{B} = \vec{R}\) (Equation 1)
- \(2\vec{A} + \vec{B}\) is perpendicular to \(\vec{B}\) (Equation 2)

2. **Understanding Perpendicular Vectors**:
- Two vectors are perpendicular if their dot product is zero. Therefore, we can write:
\[ (2\vec{A} + \vec{B}) \cdot \vec{B} = 0 \]

3. **Expanding the Dot Product**:
- Using the properties of the dot product, we expand the left-hand side:
\[ 2\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{B} = 0 \]
- Here, \(\vec{B} \cdot \vec{B} = |\vec{B}|^2\). Thus, we can rewrite the equation as:
\[ 2\vec{A} \cdot \vec{B} + |\vec{B}|^2 = 0 \]

4. **Solving for \(\vec{A} \cdot \vec{B}\)**:
- Rearranging gives us:
\[ 2\vec{A} \cdot \vec{B} = -|\vec{B}|^2 \]
- Dividing both sides by 2:
\[ \vec{A} \cdot \vec{B} = -\frac{1}{2}|\vec{B}|^2 \]

5. **Using Equation 1**:
- From Equation 1, we can express \(\vec{A}\) in terms of \(\vec{R}\) and \(\vec{B}\):
\[ \vec{A} = \vec{R} - \vec{B} \]

6. **Substituting \(\vec{A}\) into the Dot Product**:
- Substitute \(\vec{A}\) into \(\vec{A} \cdot \vec{B}\):
\[ (\vec{R} - \vec{B}) \cdot \vec{B} = -\frac{1}{2}|\vec{B}|^2 \]
- Expanding this gives:
\[ \vec{R} \cdot \vec{B} - |\vec{B}|^2 = -\frac{1}{2}|\vec{B}|^2 \]

7. **Rearranging the Equation**:
- Rearranging leads to:
\[ \vec{R} \cdot \vec{B} = \frac{1}{2}|\vec{B}|^2 \]

8. **Finding the Magnitude Relation**:
- Now, we can use the relation obtained from the dot product to find the magnitude of \(\vec{A}\):
- From the earlier derived equation:
\[ 2\vec{A} \cdot \vec{B} = -|\vec{B}|^2 \implies \vec{A} \cdot \vec{B} = -\frac{1}{2}|\vec{B}|^2 \]
- This indicates a relationship between the magnitudes of \(\vec{A}\) and \(\vec{R}\).

9. **Conclusion**:
- After analyzing the equations and substituting, we find that:
\[ |\vec{A}| = |\vec{R}| \]
- Therefore, the correct option is \( \vec{A} = \vec{R} \).